COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE
BALL QUOTIENT CASE
EDUARD LOOIJENGA
Abstract. We dene a natural compactication of an arrangement comple-
ment in a ball quotient. We show that when this complement has a moduli
space interpretation, then this compactication is often one that appears naturally by means of geometric invariant theory. We illustrate this with the
moduli spaces of smooth quartic curves and rational elliptic surfaces.
Introduction
We wish to compare two compacti cations of an algebro-geometric nature: those
obtained by means of geometric invariant theory (GIT) and the ones de ned by the
Baily-Borel theory (BB), in situations were both are naturally de ned. The examples that we have in mind are moduli spaces of varieties for which a period mapping
identi es that space with an open-dense subvariety of an arithmetic quotient of a
type IV domain or a complex ball. This includes moduli spaces of (perhaps multiply) polarized K3-surfaces, Enriques surfaces and Del Pezzo surfaces, but excludes
for instance the moduli spaces of principally polarized abelian varieties of dimension
at least three. In these examples, the GIT-boundary parametrizes degenerate forms
of varieties, so that this boundary is naturally strati ed with strata parametrizing
degenerate varieties of the same type. The strata and their incidence relations have
been computed in a number of cases, sometimes after hard work and patient analysis, but we believe that it is fair to say that these eorts failed to uncover any
well-understood, predictable pattern.
In this paper and a subsequent part we shall describe in such cases the GITcompacti cation as a stratied space in terms of the BB-compacti cation plus some
simple data of an arithmetic nature. Not only will this render the strati ed structure
more transparant, but we will also see that the theory is strong enough to be able
to guess in many cases what the GIT-compacti cation should be like.
We shall be more explicit now. Let B be a complex ball, L ! B its natural
automorphic line bundle and ; an arithmetic group operating properly on (B L).
Then the orbit space X := ;nB is a quasiprojective variety and L descends to an
`orbiline bundle' L over X. The Baily-Borel theory projectively compacti es X by
adding only nitely many points (the cusps). Suppose that we are also given a
;-invariant locally nite union of hyperballs in B . This determines a hypersurface
in X whose complement we shall denote by X . The central construction in this
paper is a natural projective compacti cation Xc of X that (i) is explicitly given
as a blowup of the Baily-Borel compactication of X followed by a blowdown and (ii)
Date : November 12, 2001.
1991 Mathematics Subject Classi cation. Primary 14J15, 32N15.
Key words and phrases. Baily-Borel compactication, arrangement, ball quotient.
1
2
EDUARD LOOIJENGA
is naturally strati ed. Perhaps the most important property for applications is that
the boundary Xc ; X is everywhere of codimension 2 in case every nonempty
intersection of the given hyperballs (including the intersection with empty index
set, i.e., B ) has dimension 2.
Here is how this is used. Let G be a reductive group acting on an integral
quasi-projective variety endowed with an ample bundle (Y ). Assume that we are
given a nonempty open subset U Y which is a union of G-stable orbits. Then
Gn(U jU) exists as a quasi-projective variety with orbiline bundle. To be precise,
the G-invariants in the algebra of sections of powers of is nitely generated and
the normalized proj of this graded algebra is a projective variety whose points correspond to the minimal orbits in the semistable part Y ss (and is therefore denoted
GnnY ss ) and contains GnU as an open-dense subvariety. Assume that we are given
an isomorphism of orbiline bundles
Gn(U jU) = (X LjX )
for some triple (X X L) as above. Assume also that the boundary at each side
(so in GnnY ss resp. Xc ) is of codimension 2. Perhaps contrary to what one might
think, there are many examples of interest for which these assumptions are satis ed
with the isomorphism in question then being given by a period map. Then we show
that the isomorphism GnU = X extends to an isomorphism
GnnY ss = Xc :
It turns out that in all cases that we are aware of the strati cation of Xc has an
interpretation in terms of the left hand side. Among these are the moduli space of
quartic curves (equivalently, of degree two Del Pezzo surfaces) and of rational elliptic brations admitting a section (equivalently, of degree one Del Pezzo surfaces).
In the course of our discussion we touch briey on a few topics that are perhaps
not indispensable for getting at our main results, but that we included here since
they involve little extra eort, can be illuminating to put things in perspective, and
have an interest on their own.
One of these starts with the observation that the construction of Xc makes sense
and is useful in a wider setting than ball quotients. For instance, we may take for
X the complement of an arrangement in a projective space or more generally,
in a torus embedding. This case is relevant for producing a compacti cation of
the universal smooth genus three curve and the universal smooth cubic surface, as
they involve adjoint tori of type E7 and E6 respectively. We showed on previous
occasions (10], 13]) that these constructions are suciently explicit for calculating
the orbifold fundamental group of these universal objects. (This enabled us to nd
a new, natural, simple presentation of the pointed mapping class group of genus
three.)
Another such topic concerns a relatively simple nontrivial necessary condition
that an `arithmetic' locally nite union of hyperballs in a complex ball must satisfy
in order that it be the zero set of an automorphic form (Section 6).
Let us nally point out that this paper builds on work of ours that goes back
to the eighties, when we set out to understand the semi-universal deformations
of triangle singularities and the GIT compacti cations of J. Shah of polarized K3
surfaces of degree 2 and 4. Our results were announced in 11], but their technical
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE 3
nature was one of the reasons that the details got only partially published (in a
preprint form in 12] and in H. Sterk's analysis of the moduli space of Enriques
surfaces 16]). We believe that the situation has now changed. Until our recent
work with Gert Heckman 7] we had not realized that the constructions also work
for ball quotients and that they form an attractive class to treat before embarking
on the more involved case of type IV domain quotients. It also turned out that
by doing this case rst helped us in nding a natural setting for the construction
(in particular, the notion 1.2 of a linearization of an arrangement thus presented
itself) which makes its relative simplicity more visible. This was for us an important
stimulus to return to these questions. In part II we shall develop the story for type
IV domains.
I thank Gert Heckman for his many useful comments on the rst version of this
paper.
1. Arrangements and their linearizations
We begin with a de nition.
Denition 1.1. Let X be a connected complex-analytic manifold and H a collection of smooth connected hypersurfaces of X. We say that (X H) is an arrangement
if each point of X admits a coordinate neighborhood such that every H 2 H meeting that neighborhood is given by a linear equation. In particular, the collection H
is locally nite.
Fix an arrangement (X H). Suppose we are given a line bundle L on X and for
every H 2 H an isomorphism L(H) OH = OH , or equivalently, an isomorphism
between the normal bundle H=X and the coherent restriction of L to H. We shall
want these isomorphisms to satisfy a normal triviality condition which roughly says
that if L is any connected component of an intersection of members of H, then these
isomorphisms trivialize the projectivized normal bundle of L. To be precise, if HL
denotes the collection of H 2 H that contain L, then the normal bundle L=X is
naturally realized as a subbundle of L OLHL by means of the homomorphism
L
L=X ! H 2HL H=X OL = L OL C C H :
The condition alluded to is given in the following
Denition 1.2. A linearization of an arrangement (X H) consists of the data of
a line bundle L on X and for every H 2 H an isomorphism L(H) OL H = OH
such that the image of the above embedding L=X ! L OL C C H is given
by a linear subspace N(L X) of C HL . (Here L is any connected component of an
intersection of members of H.) We then refer to N(L X) as the normal space of L
in X and to its projectivization P(L X) := P(N(L X)) as the projectively normal
space of L in X.
The condition imposed over L is empty when L has codimension one and is
automatically satis ed when each H is compact: if I HL consists of codimL
elements such that L is a connected component of \H 2I H, then L=X projects
isomorphically on the subsum of L C HL de ned by I and hence L=X is given
by a matrix all of whose entries are nowhere zero sections of OL . So these entries
are constant.
Here are some examples.
4
EDUARD LOOIJENGA
Example 1.3 (Ane arrangements). The most basic example is when X is a com-
plex ane space, H a locally nite collection of ane-linear hyperplanes and
L = OX . If the collection is nite, then by including the hyperplane at in nity,
this becomes a special case of:
Example 1.4 (Projective arrangements). Here X is a projective space P(W ), H
is a nite collection of projective hyperplanes and L = OPn(;1).
Example 1.5 (Toric arrangements). Now X is a principal homogeneous space of
an algebraic torus T, H is a nite collection of orbits of hypertori of T in X and
L = OX . From this example we may obtain new ones by extending these data
to certain smooth torus embeddings of X. The kernel of the exponential map
exp : Lie(T) ! T is a lattice Lie(T )(Z) in Lie(T ). So it de nes a Q-structure on
Lie(T). We recall that a normal torus embedding X X is given by a nite
collection of closed rational polyhedral cones in Lie(T)(R)g with the property
that the intersection of any two members is a common facet of these members.
The torus T acts on X and the orbits of this action are naturally indexed by
: the orbit corresponding to 2 is identi ed with the quotient T() of T by
the subtorus whose Lie algebra is the complex span of . The torus embedding is
smooth precisely when each is spanned by part of a basis of Lie(T)(Z) and in
that case := X ; X is a normal crossing divisor.
If we want the closure of H in X to meet transversally for every H 2 H,
then we must require that is closed under intersections with the real hyperplanes
Lie(TH )(R), where TH T is the stabilizer of H (a hypertorus). Then the normal
bundle of H is isomorphic to O(;) OH (choose a general T-invariant vector eld
on X and restrict to H). So O() may take the role of L.
A case of special interest is when T is the adjoint torus of a semisimple algebraic group G and H is the collection of hyperplanes de ned by the roots. The
corresponding hyperplanes in Lie(T)(R) are reection hyperplanes of the associated Weyl group and these decompose the latter into chambers. Each chamber is
spanned by a basis of Lie(T )(Z) and so the corresponding decomposition de nes
a smooth torus embedding T T . Any root of (G T), regarded as a nontrivial
character of T , extends to a morphism T ! P1 that is smooth over C . The ber
over 1 is the closure in T of the kernel of this root and hence is smooth.
Example 1.6 (Abelian arrangements). Let X be a torsor (i.e., a principal homogeneous space) over an abelian variety, H a collection of abelian subtorsors of
codimension one and L = OX .
Example 1.7 (Diagonal arrangements). Let be given a smooth curve C of genus
g and a nonempty nite set N. For every two-element subset I N the set of maps
N ! C that are constant on I is a diagonal hypersurface in C N and the collection
of these is an arrangement. But if jN j > 2 and C is a general complete connected
curve of genus 6= 1, then a linearization will not exist: it is straightforward to check
that there is no linear combination of the diagonal hypersurfaces and pull-backs
of divisors on the factors with the property that its restriction to every diagonal
hypersurface is linearly equivalent to its conormal bundle. (If the genus is one, then
the diagonal arrangement is abelian, hence linearizable.)
Example 1.8 (Complex ball arrangements). Let W be a complex vector space of
nite dimension n+1 equipped with a Hermitian form : W W ! C of signature
(1 n), B P(W) the open subset de ned (z z) > 0 and OB (;1) the restriction
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE 5
of the `tautological' bundle OP(W)(;1) to B . Then B is a complex ball of complex
dimension n. It is also the Hermitian symmetric domain of the special unitary
group SU(). The group SU() acts on the line bundle OB (;1) and the latter is
the natural automorphic line bundle over B . Every hyperplane H W of hyperbolic
signature gives a (nonempty) hyperplane section B H := P(H) \ B of B . The latter is
the symmetric domain of the special unitary group SU(H) of H. In terms intrinsic
to B : B H is a totally geodesic hypersurface of B and any such hypersurface is
of this form. Notice that the normal bundle of B H is naturally isomorphic with
OBH (1) C W=H. Hence we have an SU(H)-equivariant isomorphism
OB(;1)(B H ) OBH = OBH C W=H = OBH :
Locally nite collections of hyperplane sections of B arise naturally in an arithmetic
setting. Suppose that (W ) is de ned over an imaginary quadratic extension k
of Q, which we think of as a sub eld of C . So we are given a k-vector space
W(k) and an Hermitian form (k) : W(k) W (k) ! k such that we get (W )
after extension of scalars. We recall that a subgroup of SU()(k) is said to be
arithmetic if it is commensurable with the group of elements in SU()(k) that
stabilize the Ok -submodule spanned by a basis of W(k) (where Ok denotes the ring
of integers of k). It is known that an arithmetic subgroup of SU()(k) acts properly
discontinuously on B . Let us say that a collection H of hyperbolic hyperplanes of
W is arithmetically dened if H, regarded as a subset of P(W ), is a nite union
of orbits of an arithmetic subgroup of SU()(k) and has each point de ned over k.
In that case the corresponding collection of hyperplane sections HjB of B is locally
nite (this known fact is part of Lemma 5.4 below).
Example 1.9 (Type IV arrangements). Let V be a complex vector space of di-
mension n+2 equipped with a nondegenerate symmetric bilinear form : V V !
C . Assume that (V ) is de ned over R in such a way that has signature (2 n).
Let D P(V ) be a connected component of the subset de ned (z z) = 0 and
(z z) > 0. We let OD(;1) be the restriction of OP(V) to D as in the previous
example. Then D is a the Hermitian symmetric domain of the SL()(R)-stabilizer
of D . The group G(V ) also acts on the line bundle OD(;1) and the latter is the
natural automorphic line bundle over X. Every hyperplane H V de ned over
R of signature (2 n ; 1) gives a (nonempty) hyperplane section D H := P(HC) \ D
of D . This is a symmetric domain for the group its SL()(R)-stabilizer. As in the
previous example, it is a totally geodesic hypersurface of D , any such hypersurface
is of this form, and we have an equivariant isomorphism OH(;1)(D H ) = ODH . Any
collection of totally geodesic hypersurfaces of X that is arithmetically de ned in
the sense of the example above (with the number eld k replaced by Q) is locally
nite on D .
A ball naturally embeds in a type IV domain: if (W ) is as in the ball example,
then V := W W has signature (2 2n). A real structure is given by stipulating
that the interchange map is complex conjugation and this yields a nondegenerate
symmetric bilinear form de ned over R. For an appropriate choice of component
D , we thus get an embedding B D . If (W ) is de ned over an imaginary
quadratic eld, then (W W ) is de ned over Q and an arithmetic arrangement
on D restricts to one on B .
6
EDUARD LOOIJENGA
2. Blowing up arrangements
2.1. Blowing up a fractional ideal. In this subsection we briey recall the basic
notion of blowing up a fractional ideal. Our chief references are 6] and 5].
k
Suppose X is a variety and I OX is a coherent ideal. Then 1
k=0 I is a OX algebra that is generated as such by its degree one summand (it should be clear
that I 0 := OX ). Its proj is a projective scheme over X, : X~ ! X, called the
blowup of I , with the property that IX is invertible and relatively very ample.
If X is normal and I de nes a reduced subscheme of X, then its blowup is
normal also. The variety underlying X~ is over X locally given as follows: if I is
generated over an open subset U X by f0 : : : fr 2 C U], then these generators
determine a rational map f0 : : fr ] : U ! Pr and the closure of the graph of
this map in U Pr with its projection onto U can be identi ed with (X~U )red ! U.
This construction only depends on I as a OX -module. So if L is an invertible sheaf
on X, and I is a nowhere zero coherent subsheaf of L, then we still have de ned
the blowup of I as the proj of 1
I (k) 1k=0Lk, where I (k) denotes the kth
k=0
k
k
power of I : the image of I in L . In fact, for the de nition it suces that I is
a nowhere zero coherent subsheaf of the sheaf of rational sections of L.
The coherent pull-back of I along X~ ! X is invertible and the latter morphism
is universal for that property: any morphism from a scheme to X for which coherent
~ In particular, if Y X is a closed
pull-back of I is invertible factorizes over X.
~
subvariety, then the blowup X ! X of the ideal de ning Y is an isomorphism
when Y is a Cartier divisor. If Y is the support of an eective Cartier divisor, then
X~ ! X is still nite, but if Y is only a hypersurface, then some bers of X~ ! X
may have positive dimension.
Here is a simple example that has some relevance to what will follow. The
fractional ideal in the quotient eld of C z1 z2] generated by z1;1z2;1 is uninteresting
as it de nes the trivial blowup of C 2 . But the blowup of the ideal generated by z1;1
and z2;1 amounts to the usual blowup of the origin in C 2 .
2.2. The arrangement blowup. Let (X H) be an arrangement. There is a simple and straightforward way to nd a modi cation XH ! X of X such that the
preimage of D is a normal crossing divisor: rst blow up the union of the dimension
0 intersections of members of the H, then then the strict transform of the dimension
1 intersections of members of the H, and so on, nishing with blowing up the strict
transform of the dimension n ; 2 intersections of members of the H:
X = X 0 X 1 X n;2 = X~ H :
We refer to X~ H as the blow-up of X de ned by the arrangement H or briey, as
the H-blow-up of X. To understand the full picture, it is perhaps best to do one
blow-up at a time. In what follows we assume that we are also given a linearization
L of the arrangement (X H). We begin with a basic lemma.
Let us denote by PO(H) the partially ordered set of connected components of
intersections of members of H which have positive codimension. For L 2 PO(H),
we denote by HL the collection of H 2 H containing L as a lower dimensional
subvariety (so this collection is empty when codim(L) = 1). The following lemma
is clear.
Lemma 2.1. Given L 2 PO(H), then the members of HL dene a linear arrangement in N(L X). If L0 2 PO(H) contains L, then we have a natural exact sequence
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE 7
of vector spaces
0 ! N(L L0 ) ! N(L X) ! N(L0 X) ! 0:
P If L is an invertible sheaf on X, then we shall write L(H) for the subsheaf
H 2H L(H) of the sheaf of rational sections of L. This sheaf is a coherent OX module, but need not P
be invertible. In particular, it should not be confused with
the invertible sheaf L( H 2H H). The kth power of L(H) `as a fractional ideal' is
L(H)(k) := X Lk(H1 + + Hk)
(H1 :::Hk )2Hk
(k)
and so its blowup is the proj of 1
k=0 ( ) .
LH
Let L X be a minimal member of PO(H) and of codimension 2 and let
: X~ ! X blow up L. The exceptional divisor E := ;1L is then the projectivized
normal bundle P(L=X ) of L. In case a linearization exists, the previous lemma
identi es this exceptional divisor with L P(L X) in such a manner that the strict
transform H~ of H 2 HL meets it in L P(L H).
Lemma 2.2. Suppose that the arrangement (X H) is linearized by the invertible
sheaf L. Then the strict transform of H in X~ is an arrangement that is naturally
linearized by L~ := L(E). Precisely, if stands for the exterior coherent tensor
product, then
(i) E=X~ = (L OL ) OP(LX)(;P1),
(ii) L(H) OE = (L OL ) ( H 2HL OP(LX)(;1)(P(L H)) and
~ OH~ (iii) for every H 2 HL we have L~(H)
= OH~ .
Proof. Assertion (i) is an immediate consequence of Lemma 2.1. If H 2 HL , then
~ = ( L)(E + H)
~ = (L(H)) to the preimage of
the coherent restriction of L~(H)
~ which
H is trivial. Hence the same is true for its coherent restriction to E or H,
~
proves assertion (ii). It is clear that OX~ (H) OE = OL OP(LX)(P(L H)). Now
(iii) follows also:
X
~
L(H) OE = ( L)(
OE (E + H))
H 2HL
X
~
E=X~ (
E (E H)
L
H 2H
X
L) (
P
(LX)( 1)(P(L H)):
H 2HL
L
= (L O
=
O \
O ;
Hence (X~ fH~ gH 2H L~) satis es the same hypotheses as (X fH gH 2H L). Our
gain is that we have eliminated an intersection component of H of minimal dimension. We continue in this manner, until the strict transforms of the members of H
are disjoint and end up with X~ H ! X.
Convention 2.3. If an arrangement H on a connected complex manifold X is
understood, we often omit it from notation that a priori depends on H. For instance, we may write X~ for the corresponding blow-up X~ H of X and X for the
~
complementary Zariski open subset (as lying either in X or in X).
8
EDUARD LOOIJENGA
The members of H ; HL that meet L de ne an arrangement HL in L. So we
have de ned L~ and L . It is realized as the strict transform of L under the blowups
of members of PO(H) smaller than L. Lemma 2.2 shows that the projectivized
normal bundle of L~ in X can be identi ed with the trivial bundle L~ P(L X) such
that the members of HL determine a projective arrangement in P(L X). So we
~ (L X) and P(L X) . The preimage of L~ in X~ is a smooth divisor
have de ned P
that can be identi ed with
~ (L X):
E(L) := L~ P
It contains
E(L) := L P(L X) :
as an open-dense subset.
Lemma 2.2 yields with induction:
Lemma 2.4. Under the assumtions of Lemma 2.2 we have:
(i) The morphism : X~ ! X is obtained by blowing up the fractional ideal sheaf
O(H),
(ii) for every L 2 PO(H), the coherent pull-back of L(H) to E(L) is identied
with the coherent pull-back of the sheaf OP(LX)(;1)(HL ) on P(L X) under
~ (L X) ! P(L X) and
the projection E(L) ! P
(iii) the coherent pull-back of L(H) to L~ is identied with the coherent pull-back
of L(HL ) to L~ (a line bundle by (i)) tensorized over C with the vector space
H 0(OP(LX)(;1)(HL )).
Lemma 2.5. If L and L0 are distinct members of PO(H), then E(L) and E(L0 )
do not intersect unless L and L0 are incident. If they are, and L L0 , say, then
~ (L L0 ) P
~ (L0 X):
E(L) \ E(L0 ) = L~ P
Proof. The rst assertion is clear. The strict transform of L in the blowing ups
of members of PO(H) contained in L is L~ P(L X). The strict transform of L0
meets the latter in L~ P(L L0). Then E(L) \ E(L0 ) must be the closure of the
~ If we next blow up the members of PO(H) strictly
preimage of L~ P(L L0) in X.
0
~ (L L0 ). Blowing up L0
between L and L , the strict transform of L~ P(L L0) is L~ P
0
0
~ (L L ) P(L X), Finally, blowing up the members of PO(H) strictly
yields L~ P
~ (L L0 ) P
~ (L0 X). The lemma now follows easily.
containing L0 gives L~ P
This generalizes in a straightforward manner as follows:
Lemma 2.6. Let L0 L1 : : : Lr be distinct members of PO(H), ordered by dimension: dim(L0 ) dim(L1 ) dim(Lr ). Then \i E(Li ) is nonempty if and only
if these members make up a ag L = (L0 L1 Lr ) and in that case the
intersection in question equals
~ (L0 L1) P
~ (Lr;1 Lr ) P
~ (Lr X):
E(L ) := L~ 0 P
Moreover, its open-dense subset
E(L ) := L0 P(L0 L1)
P(Lr;1 Lr ) P(Lr X)
is a minimal member of the Boolean algebra of subsets of XH generated by the
divisors E(L). These elements dene a stratication of X~ ; X .
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE 9
2.3. Minimal (Fulton-MacPherson) blowup of an arrangement. The simple blowing-up procedure described above turns the arrangement into a normal
crossing divisor, but it is clearly not minimal with respect that property. If V is a
vector space and (Wi V )i is a collection subspaces that is linearly independent in
the sense that V ! i V=Wi is surjective, then the blowup of iWi de nes a morphism V~ ! V for which the total transform of i Wi is a normal crossing divisor:
the strict transform W~ i of Wi is a smooth divisor and these divisors meet
Q transversally, their common intersection being bered over \iWi with ber i P(V=Wi).
It is also obtained by blowing up the strict transforms of the Wi 's in any order.
So in case of an arrangement we can omit the blowups along L 2 PO(H) with the
property that the minimal elements of HL are independent along L in the above
sense. Since we still end up with a normal crossing situation, we call this the minimal normal crossing resolution of the arrangement. It is not hard to specify a
fractional ideal on X whose blowup yields this minimal normal crossing resolution.
These and similar blowups have been introduced and studied by Yi Hu 8] who
built on earlier work of A. Ulyanov 17].
A case that is perhaps familiar is the diagonal arrangement of Example 1.7: we
get the Fulton-MacPherson compacti cation of the space of injective maps from a
given nite nonempty set to a given smooth curve C. (We may subsequently pass
to the orbit space relative to the action automorphism group of C if that group
acts properly.)
3. Contraction of blown-up arrangements
Throughout the rest of this paper, (X H L) stands for a linearized arrangement.
According to Lemma 2.4 the pull-back L(H) on X~ is invertible. Suppose for
a moment that X is compact and L(H) is generated by its sections. Then so are X~
and L(H) and hence the latter de nes a morphism from X~ to a projective space.
By Lemma 2.4 again, this morphism will be constant on the bers of the projections
~ (L X). In other words, if R is the equivalence relation on X~ generated
E(L) ! P
by these projections, then the morphism will factorize through the quotient space
~
X=R.
Our goal is to nd conditions under which L(H) is semiample, more
precisely, conditions under which this quotient space is projective and L(H) is
~
the coherent pull-back of an ample line bundle on X=R.
Let us rst focus on R as an equivalence relation. We begin with noting that on
a stratum E(L0 Lr ) the equivalence relation is de ned by the projection
~ is as a set the disjoint union of
on the last factor P(Lr X) . So the quotient X=R
X and the projective arrangement complements P(L X) , L 2 PO(H).
Lemma 3.1. The equivalence relation R, when viewed as a subset of X~ X~ , is
equal to the union of the diagonal and the E(L) P
~(LX) E(L). In particular, R is
~
closed and hence X=R is Hausdor.
Proof. By de nition R contains the union in the statement. The opposite inclusion
also holds: x 2 E(L0 Lr ) and x0 2 E(L00 L0s ) are related if and
only if Lr = L0s and both have the same image in P(Lr X) .
Theorem 3.2. Suppose X compact and that some positive power of L(H) is generated by its sections and that these sections separate the points of X . Then the
pull-back of L(H) to X~ is a semi-ample invertible sheaf and a positive power of this
10
EDUARD LOOIJENGA
sheaf denes a morphism from X~ to a projective space whose image X^ realizes the
quotient space X=R with the strata X and P(L X) of X=R realized as subvarieties
of X^ .
Proof. Let L(H)(k) be generated by its sections and separate the points of X .
Then the invertible sheaf (L(H)(k) ) has the same property. Let f : X~ ! PN
be the morphism de ned by a basis of its sections. A possibly higher power of
L(H) will have a normal variety as its image and so it suces to prove that f
~
separates the points of every stratum of X=R.
By hypothesis this is the case for
X . Given L 2 PO(H) of codimension 2, then it follows from Lemma 2.4 that
the pull-back of L(H) to L~ is isomorphic to a line bundle on L~ tensorized over C
with H 0 (OP(LX)(;1)(HL )). Any set of generating sections of L(H) must therefore
determine a set of generators of the vector space H 0 (OP(LX)(;1)(HL )). Such a
generating set separates the points of P(L X) . The same is true for a set of
generating sections of a power of L(H). So for some positive k, L(H)(k) separates
~
the points of every stratum of X=R.
This theorem applies to some of the examples mentioned at the beginning.
3.1. Projective arrangements. The hypotheses of Theorem 3.2 are satis ed if
the collection H of hyperplanes has no point in common: if fH 2 V is a de ning
equation for H P(V ), then OP(V) (;1)(H) is generated by the sections ffH;1 gH 2H .
^ (V ) appears
There is no need to pass to a higher power of this sheaf: the variety P
H
as the image of the rational map P(V ) ! P(C ) with coordinates (fH;1 )H 2H . If
V = C n+1 and H is the set of coordinate hyperplanes, then P^n = Pn and this map
is just the standard Cremona transformation in dimension n.
Of particular interest are the cases when H is the set of reection hyperplanes
^ (V ) has a G-action
of a nite complex reection group G with V G = f0g. Then P
^ (V ). (A theorem of Chevalley implies that
so that we can form the orbit space GnP
GnP(V) is rational, hence so is this completion.)
3.2. Toric arrangements dened by root systems. Let T be the adjoint torus
of a semisimple algebraic group. It comes with an action of the Weyl group W. The
collection of reection hyperplanes of W in Lie(T)(R) de nes a torus embedding
T T . The roots de ne a collection of `hypertorus embeddings' H in T . We
may also state this as follows: the set R of roots de ne an embedding T ! (C )R ,
T is the closure of its image in (P1)R and H is the ber over 1 : 1] of the
projection of T ! P1 on the factor indexed by . A T-orbit T() in T T
is also an adjoint torus (associated to a parabolic subgroup), its closure in T
is the associated torus embedding (hence smooth) and the roots of T restrict to
roots of T () and vice versa. Denote by the collection of indivisible elements of
Lie(T )(Z) that generate a (one dimensional) face of (these are just the coweights
that are fundamental relative a chamber). The closure D($) of T(R>0$) in T
is a (smooth) T -invariant divisor and all such are so obtained. The meromorphic
functions ff := (e ; 1);1g2R separate the points of T . They also separate the
T -orbits from each other: The divisor of f is
;H + X ($)D($):
21
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE11
Two distinct members of can be distinguished by a root and so the corresponding
T-orbits are separated by the corresponding meromorphic function. Given a $ 2 ,
then each f with ($) = 0 restricts to a meromorphic function on D($) of the
same type. For that reason these meromorphic functions also separate the points
of any stratum T() .
So Tc exists as a projective variety. It comes with an action of the Weyl group W
of the root system. Notice that the subvariety of Tc over the identity element of T
^ (Lie(T)) of the projectivized tangent space P(Lie(T )) de ned
is the modi cation P
by the arrangement of tangent spaces of root kernels.
Example 3.3 (The universal cubic surface). Suppose (T W ) is an adjoint torus
of type E6. Then the construction of 10] (see also 13]) shows that (f1g W)nTc
may be regarded as a compacti cation of the universal smooth cubic surface. Let us
briey recall the construction. If S P3 is a cubic surface, then for a general p 2 S,
the projective tangent space of S at p meets S in a nodal cubic Cp . The latter is also
an anticanonical curve. The identity component Pico (Cp ) of the Picard group of
Cp is a copy of C . The preimage Pico (S) of Pico (Cp ) in Pic(S) is a root lattice of
type E6 and so Hom(Pico (S) Pico (Cp )) is an adjoint E6 -torus. The restriction map
de nes an element of this torus. It turns out that it does not lie in any reection
hypertorus. We proved in 10] that this element is a complete invariant of the pair
(S p): the f1g W -orbit in T de ned by the restriction map determines (S p) up
to isomorphism. All orbits in T so arise and so (f1g W )nT is a coarse moduli
space for pairs (S p) as above.
If we allow p to be an arbitrary point of a smooth cubic, then Cp may degenerate
into any reduced cubic curve. The type of this curve corresponds in fact to f1gW invariant union of strata of Tc in a way that (f1g W)nT is going to contain the
coarse moduli space of pairs (S p) with p an arbitrary point of the smooth cubic
S. To be precise: if Cp becomes a cuspidal curve (with a cusp at p), then there is
a unique stratum, it is projective of type E6 (it is the one de ned by the identity
element of T: a copy of P(Lie(T )) ). If Cp becomes a conic plus a line, then we get
the projective or toric strata of type D5 , depending on whether or not the line is
tangent to the conic. Finally, if Cp becomes a sum of three distinct lines, then we
get projective or toric strata of type D4 , depending on whether or not the lines are
concurrent.
Example 3.4 (The universal quartic curve). The story is quite similar to the preceding case (we refer again to 10] and 13]). We now assume that (T W) is an
adjoint torus of type E7. A smooth quartic curve Q in P2 determines a Del Pezzo
surface of degree 2: the double cover S ! P2 rami ed along Q. This sets up a bijective correspondence between isomorphism classes of quartic curves and isomorphism
classes of Del Pezzo surface of degree 2. If p 2 Q is such that the projective line
Tp Q meets Q in two other distinct points, then the preimage Qp of Tp Q in S is a
nodal genus one curve. Starting with the homomorphism Pic(S) ! Pic(Qp ), we
de ne Pico (S) as before (it is a root lattice of type E7) so that we have an adjoint
E7-torus Hom(Pico (S) Pico (Qp )). Proceeding as in the previous case we nd that
W nT is a coarse moduli space for pairs (Q p) as above (now ;1 2 W ) and that
allowing p to be arbitrary yields an open-dense embedding of the corresponding
coarse moduli space in W nTc . The added strata are as follows: allowing Tp Q to
become a ex (but not a hyperex) point yields the projective stratum of type E7
12
EDUARD LOOIJENGA
over the identity element. If Tp Q is a bitangent, then we get the toric or projective
strata of type E6, depending on whether the bitangent is a hyperex. Our compacti cation of W nT has another interesting feature as well. We have also A6 and
A7 -strata, both of projective type. These are single orbits and have an interpretation as the coarse moduli space of pairs (Q p), where p is a point of a hyperelliptic
curve Q of genus 3: we are on a A6 or A7-stratum depending on whether or not p is
a Weierstrass point. So W nTc contains in fact the coarse moduli space of smooth
pointed genus three curves M31. We used aspects of this construction in 11] and
13] to compute the rational cohomology resp. the orbifold fundamental group of
M31.
3.3. Abelian arrangements. Let X be a torsor over an abelian variety A, and
H a nite collection of abelian subtorsors of codimension one. Denote by A0 the
identity component of the group of translations of X that stabilize H (an abelian
variety).
First assume that A0 reduces to the identity element. This ensures that
OX (PH2H H) is ample. Hence a power of OX (H) separates the points of X . It
is now easy to see that X^ is de ned. For every H 2 H, X=H is an elliptic curve
(the origin is the image of H). A rational function on X=H that is regular outside
the origin and has a pole of order k at the origin can be regarded as a section
of OX (kH). For a suciently large k, the collection of these functions de ne a
^ The morphism
morphism from X~ to a projective space which factorizes over X.
^
from X to this projective space is nite. In the general situation (where A0 may
have positive dimension), the preceding construction can be carried out in a A0equivariant manner to produce a projective completion X^ of Y with A0 -action.
Here is a concrete example. If R is a reduced root system with root lattice Q
and E is an elliptic curve, then X := Hom(Q E) is an abelian variety on which
the Weyl group W of R acts. The xed point loci of reections in W de ne an
abelian arrangement H on X with \H 2H H nite. So X^ is de ned and comes with
an action of W.
Example 3.5. For R of type E6 , (f1g W)nX is the moduli space of smooth
cubic surfaces with hyperplane section isomorphic to E, provided that E has no
exceptional automorphisms. Hence (f1g W)nX^ is some compacti cation of this
moduli space.
Example 3.6. For R of type E7 there is a similar relationship with the moduli
space of smooth quartic curves with a line section for which the four intersection
points de ne a curve isomorphic to E.
3.4. Ample line bundles over abelian arrangements. This is a variation on
3.3, which we mention here because of a later application to automorphic forms
on ball quotients and type IV domains with product expansions. Suppose that in
the situation of 3.3 we are given an ample line bundle ` over X. Let C be the
corresponding ane cone over X, that is, the variety obtained from the total space
of the dual of ` by collapsing its zero section. In more algebraic terms, CH is spec of
0 k
the graded algebra 1
k=0H (` ). Each H 2 H de nes a hypersurface (a subcone
of codimension one) CH C. We put CH := H 2H CH .
Lemma 3.7. If H 6= , then the hypersurface CH is the zero set of a regular function on C if and only if the class of ` in Pic(X) Q is a positive linear combination
of the classes H , H 2 H.
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE13
Proof. If f is a regular function on C with zero set CH , then f mustPbe homoge-
neous, say of degree k. So f then de nes a section of ` P
with divisor H 2H nH H,
with nH a positive integer. This means that `k O
(
= X H nH H). Conversely, an
identity of this type implies the existence of such an f.
Since ` is ample, the condition that ` is a positive linear combination of the
classes indexed by H, is not ful lled if A0 (the identity component of the group of
translations of X that stabilize H) is positive dimensional.
3.5. Dropping the condition of smoothness. If in the situation of the previous theorem, Y is a normal subvariety of X which meets the members of PO(H)
transversally, the strict transform of Y in X^ only depends on the restriction of the
arrangement to Y . This suggests that the smoothness condition imposed on X can
be weakened to normality. This is indeed the case:
Denition 3.8. Let X be a normal connected (hence irreducible) variety, L be a
line bundle over X. A linearized arrangement on X consists of a locally nite collection of (reduced) Cartier divisors fH gH 2H on X, a line bundle L and and for each
H 2 H an isomorphism L(H) OH = OH such that for every connected component
L of an intersection of members of H the following conditions are satis ed:
(i) the natural homomorphism H 2HL IH =IH2 OL ! IL=IL2 is surjective and
not identically zero on any summand,
(ii) if we identify H 2HL IH =IH2 OL with LOL C C HL via the given isomorphisms, then the
kernel of the above homomorphism is spanned by a subspace
K(L X) C HL whose codimension is that of L in X.
Notice that these conditions imply that the conormal sheaf IL =IL2 is a locally free
OL-module of rank equal to the codimension of LHLin X. The normal space N(L X)
is now de ned as the space of linear forms on C that vanish on K(L X).
The conditions of this de nition are of course chosen in such a manner as to
ensure that the validity of the discussion for the smooth case is not aected. In
particular, we have de ned a blow-up X~ ! X whose exceptional locus is a union
~ (L X).
of divisors E(L) = L~ P
Theorem 3.9. Suppose that is X compact and that a power of L(H) is generated
by its sections and separates the points of X . Then the pull-back of L(H) to
X~ is semi-ample and the corresponding projective contraction X~ ! X^ has the
property that X^ ; X is naturally stratied into subvarieties indexed by PO(H):
to L 2 PO(H) corresponds a copy of P(L X) (this assignment reverses the order
relation).
4. Complex ball arrangements
We take up Example 1.8. We shall assume that n 2. It is clear that Theorem
3.2 does not cover this situation and so there is work to do.
4.1. The ball and its natural automorphic bundle. We choose an arithmetic
subgroup ; of SU()(k), although we do not really want to x it in the sense that
we always allow the passage to an arithmetic subgroup of ; of nite index. In
particular, we shall assume that ; is neat, which means that the multiplicative
subgroup of C generated by the eigen values of elements of ; is torsion free. This
implies that every subquotient of ; that is `arithmetically de ned' is torsion free.
14
EDUARD LOOIJENGA
We write X for the ;-orbit space of B . A cusp of B relative the given k-structure
is an element in the boundary of B in P(W) that is de ned over k. A cusp corresponds to an isotropic line I W de ned over k.
Denote by L W the set of w 2 W with (w w) > 0. The obvious projection
L ! B is a C bundle. We may view this as the complement of the zero section
of the tautological line bundle over P(W) restricted to B :
L := L C C :
A a nonzero v 2 L de nes a homogeneous function f : L ; f0g ! C of degree ;1
characterized by the property that f(v) = 1. So a nonzero holomorphic function
on L that is homogeneous of degree ;k de nes a holomorphic section of Lk and
vice versa.
Let us point out here the relation between L and the canonical bundle of B . If
p 2 B is represented by the line L W, then the tangent space Tp B is naturally
isomorphic to Hom(L W=L). So ^nTp B = Ln+1 ^n+1 W . This proves that the
canonical bundle of B is SU()-equivariantly isomorphic to L(n+1). We regard L
as the natural automorphic bundle over B . Since ; is neat, L drops to a line bundle
over X.
4.2. The stabilizer of a cusp. The following bit of notation will be useful.
Convention 4.1. Given a subspace J of W and a subset W resp. P(W) ;
P(J), then (J) denotes the image of in W=V resp. P(W=J).
Suppose that J is a degenerate subspace of B , so that I := J \ J ? is an isotropic
line with I ? = J + J ? . Then
L(J) = W=J ; I ? =J and hence B (J) = P(W=J) ; P(I ?=J):
In the rst case we have the complement of a linear hyperplane in a vector space!
in the second case the complement of a projective hyperplane in a projective space,
hence an ane space (with Hom(W=I ? I ? =J) as translation group). The former
is a C -bundle over this ane space. Notice that for the maximal choice J = I ? ,
this is a C -bundle over a singleton.
Let I W be an isotropic line. We begin with recalling the structure of the
stabilizer SU()I . If e is a generator of I, then the unipotent radical NI of the
SU()-stabilizer of I consists of transformations of the form
Tev : z 2 W 7! z + (z e)v ; (z v)e ; 12 (v v)(z e)e
for some v 2 I ? . Notice that Tev = Tev and that Tev+e = Tev when 2 R.
So Tev only depends on the image of e v in I C I ? =(I I)(R) (observe I I
has a natural real structure indeed). A simple calculation yields
TeuTev = Teu+v+ 21 (uv)e
showing
p that NI is an Heisenberg group: its center Z(NI ) is identi ed with the real
line ;1(I C I)(R) and the quotient NI =Z(NI ) with the vector group I I ? =I,
the commutator given essentially by the skew form that the imaginary part induces on I ? =I. One can see this group act on B as follows. The domain B lies
in P(W) ; P(I ?). The latter is an ane space for Hom(W=I ? I ? ) and so NI will
act on this as an ane transformation group. This group preserves the bration
P(W) ; P(I ?) ! P(W=I) ; P(I ?=I) = B (I):
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE15
This is a bration by ane lines whose base P(W=I) ; P(I ?=I) is an ane space
over Hom(W=I ? I ? =I). The center Z(NI ) of NI respects each ber and acts there
as a group of translations and the quotient vector group NI =Z(NI ) is faithfully
realized on the base as its group of translations. The domain B is bered over B (I)
by half planes such that the Z(NI )-orbit space of a ber is a half line.
Let us rst decribe B as such in terms of (partial) coordinates: choose f 2 V
such that (e f) = 1 and denote the orthogonal complement of the span of e and
f by A. It is clear that A is negative de nite and can as an inner product space be
identi ed with I ? =I. The map (s a) 2 C A 7! se + f + a] 2 P(V ) ; P(I ?) is an
isomorphism of ane spaces and in these terms B is de ned by 2Re(s) > ;(a a).
This is known as the realization of B as a Siegel domain of the second kind.
A somewhat more intrinsic description goes as follows. Consider the group homomorphism Te : C ! GL(W) de ned by
Tes (z) = z + s(z e)e:
A simple computation shows that
(Tes z Tesz) = (z z) + 2Re(s)j(z e)j2 :
The restriction of Te to the imaginary axis parametrizes Z(NI ) and therefore Te
maps to the complexi cation SU(). Since T s maps B into itself for Re(s) 0, the
image TI in GL(W) of the right half plane is a semigroup that acts (freely) on B .
As the notation suggests it does not depend on the generator e of I. The bers of
B ! B (I) are orbits of this semigroup. The image of T lies in the complexi cation
of SU(). It is normalized by NI and so NI TI = TI NI is a semigroup in SU()C .
The domain B is a free orbit of this semigroup.
Suppose now that I is de ned over k. The Levi quotient SU()I =NI of SU()I is
the group of unitary transformations of I ? =I, hence compact, and the corresponding subquotient of ; is nite. Since we assumed ; to be neat, the latter is trivial
and so the stabilizer ;I is contained in NI . It is in fact a cocompact
subgroup of
p
NI : the center Z(;I ) of ;I is in nite cyclic (a subgroup of ;1(I I)(R)), and
;I =Z(;I ) is an abelian group that naturally lies as a lattice in the vector group
I I ? =I.
4.3. Compactications of a ball quotient. Let J be a complex subspace ofW
with radical I. The group N J of 2 ; that act as the identity on J ? = W=J is
a normal (Heisenberg) subgroup of NI with quotient identifyable with the vector
group I I ? =J). The latter may be regarded as the vector space of translations of
the ane space B (J) = P(W=J) ; P(I ?=J). Let us assume that J is de ned over
k. Then the discrete counterparts of these statements hold for ;. In particular,
;I =;J can be identi ed with a lattice in I J=I. So if we denote the orbit space of
this lattice acting on B (J) by X(J), then X(J) is a principal homogenenous space
of a complex torus that has I I ? =J) as its universal cover. For an intermediate
k-space I J 0 J we have a natural projection X(I) ! X(J) of abelian torsors.
Notice that X(I ? ) is just a singleton.
Denition 4.2. An arithmetic system (of degenerate subspaces) assigns in a ;equivariant manner to every k-isotropic line I a degenerate subspace j(I) de ned
over k with radical I. We call the two extremal cases bb(I) := I ? resp. tor(I) := I
the Baily-Borel system and the toroidal system respectively.
16
EDUARD LOOIJENGA
Such a system j leads to a compacti cation X j of X as follows. Form the disjoint
unions
a
a
(L)j := L t
L(j(I)) and B j := B t
B (j(I)):
I k-isotropic
I k-isotropic
Both come with an obvious action of ; and the projection (L)j
! B j is a C bundle. We introduce a ;-invariant topology on these sets as follows. Recall that
N j(I) is the subgroup of g 2 NI that act as the identity on W=j(I). Then N j(I) TI =
TI N j(I) is a semigroup and thej bers of B ! B (j(I)) are orbits of this semigroup.
A basis of a topology on (L ) is speci ed by the following collection of subsets:
(i) the open subsets of L,
(ii) for every k-isotropic line I and N j(I) TI -invariant open subset L the
subset t (j(I)).
The same de nition with obvious modi cations de nes a topology for B j . It makes
Lj := (L)j C C :
a topological complex line bundle over B j . We are interested in the orbit space
X j := ;nB j
and the line bundle over X j de ned by ;nLj. Notice that as a set, X j is the disjoint
union of X and the torsors X(j(I)), where I runs over a system of representatives of
the ;-orbits in the set of cusps. Fundamental results from the theory of automorphic
forms assert that:
(i) ; has only nitely many orbits in its set of cusps and X j is a compact Hausdor space.
(ii) The sheaf of complex valued continuous functions on X j that have analytic
restrictions to the strata X and X(j(I)) gives X j the structure of a normal
analytic variety.
(iii) The line bundle Lj ! B j descends to an analytic line bundle on ;nB j (a local
section is analytic if it is continuous and analytic on strata). We denote its
sheaf of sections by L (the suppression of j in the notation is justi ed by (v)
below).
(iv) The line bundle L is ample on X bb, so that the graded algebra
A := k H 0 (X bb Lk)
is nitely generated with positive degree generators, having X bb as its proj.
re nes j in the sense that j 0 (I) j(I) for
(v) If j 0 is an arithmetic system that
0
all I, then the obvious map
X j ! X j is a morphism of analytic spaces and
0
j
the line bundle L on X is the pull-back of of its namesake on X j .
The projective variety X bb is known as the Baily-Borel compactication and X tor
as the toric compacti cation of X. The boundary of X bb ; X is nite (its points
are called the cusps of X bb ).
Let us see how this works out locally. Given a k-isotropic line, then the orbit
space EI := ;I n(P(W) ; P(I ?)) lies as a C -bundle over X(I) and contains ;I nB
as a punctured disc bundle. The obvious morphism ;I nB ! ;nB = X restricts
to an isomorphism of an open subset of X (in fact, a punctured neighborhood of
the cusp de ned by I) onto a smaller punctured disc bundle so that the insertion
of X(j(I)) can be described in terms of the C -bundle EI ! X(I). For instance,
the toric compacti cation amounts to simply lling in the zero section: we get EItor
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE17
as a union of EI and X(I). The line bundle EItor ! X(I) is anti-ample (the real
part of I is a Riemann form). This implies that we can contract the zero section
analytically! the result is a partial Baily-Borel compacti cation EIbb, which adds to
E I a singleton. The intermediate case EIj (a union of EI and X(j(I))) is obtained
from EItor by analytical contraction via the map X(I) ! X(j(I)). (This in indeed
possible.)
5. Arithmetic ball arrangements and their automorphic forms
Let now be given a collection H of hyperplanes of W of hyperbolic signature that
is arithmetically de ned relative to ;. So H, when viewed as a subset of P(W ), is
a nite union of ;-orbits and contained in P(W )(k).
We note that for each H 2 H, B H = P(H) \ B is a symmetric subdomain of
B . Its ;-stabilizer ;H is arithmetic in its SU()-stabilizer and so the orbit space
XH := ;H nB H has its own Baily-Borel compacti cation XHbb . The inclusion B H B
induces an analytic morphismXH ! X. This morphismis nite and birational onto
its image (which we denote by DH ) and extends to a nite morphism XHbb ! X bb
whose image is the closure of DH in X bb (which we denote by DHbb). Clearly, DHbb
only depends on the ;-equivalence class of H and so the union DHbb := H 2H DHbb is
a nite one.
Lemma 5.1. After passing to an arithmetic subgroup of ; of nite index, each
hypersurface DHbb is without self-intersection in the sense that XHbb ! DHbb is a
homeomorphism and XH ! DH is an isomorphism.
Proof. We do this in two steps. The set SH of 2 ; such that B H \B H is nonempty
and not equal to B H is a union of double cosets of ;H in ; distinct from ;H . Each
such double coset de nes an ordered pair of distinct irreducible hypersurfaces in
XH with the same image in X and vice versa. Since XH ! DH is nite and
birational, there are only nitely many such pairs of irreducible hypersurfaces and
hence SH = ;H CH ;H for some nite set CH ;H . Let C be the union of the sets
CH , for which H runs over a system of representatives of ; in H. This is a nite
subset of ; ; f1g. If we now pass from ; to a normal arithmetic subgroup of ;
of nite index which avoids C, then the natural maps XH ! X are injective and
local embeddings. They are also proper, and so they are closed embeddings.
It is still possible that XHbb ! X bb fails to be injective for some for some H
(although this can only happen when n = 2). We avoid this situation essentially
by proceeding as before: given isotropic line I W de ning a cusp then the set
2 ; ; ;I such that for some H 2 H H 6= H and H \ H I is a union of
double cosets of ;I in ;. The number of such double cosets is at most the number
of branches of DHbb at this cusp and hence nite. If CI0 ; ; ;I is a system of
representatives, then let C 0 be a of union CI0 ', where I runs over a ( nite) system
of representatives of ; in the set B bb ; B . Choose a normal arithmetic subgroup of
; of nite index which avoids C 0 . Then this subgroup is as desired.
We assume from now on that the hypersurfaces DHbb are without self-intersection
in the sense of Lemma 5.1 and we identify XH with its image DH in X. Then
the collection H; of the hypersurfaces DHbb is an arrangement on X. We want to
determine whether the algebra
0 bb
(k)
AH := 1
k=0 H (X L(H;) ))
18
EDUARD LOOIJENGA
is nitely generated and if so, nd its proj.
The hypersurface DHbb on X bb will in general not support a Cartier divisor. On
the other hand, the closure DHtor of DH in X tor is even smooth. So the blow-up of
DHbb in X bb will be dominated by X tor . This is then also true for the minimalnormal
blow-up of X bb for which the strict transform of each DHbb is a Cartier divisor. It is
easy to describe this intermediate blow-up of X bb . For this, we let for a k-isotropic
line I W, jH (I) be the common intersection of I ? and the members of H that
contain I.
Lemma 5.2. The assignment jH is an arithmetic system relative to ; and the
corresponding morphism X jH ! X bb is the smallest normalized blow-up that makes
the fractional ideals OX bb (DHbb ) invertible. The strict transform of DHbb in X jH is
an integral Cartier divisor.
Proof. Let I W be a k-isotropic line. Then J := jH (I) contains I, is contained
in I ? and is de ned over k since it is an intersection of hyperplanes de ned over k.
bb in E bb and an abelian
Let H 2 H contain I. Then H de nes a hypersurface EIH
I
subtorsor X(I)H of X(I) with quotient X(H \ I ? ). The latter is a genus one curve
which apparently comes with an origin o (it is an elliptic curve) and the complement
of this origin of is ane. Any regular function on the ane curve X(H \ I ? ) ;fog
tor with the same
with given pole order at o lifts to a regular function on EItor ; EIH
tor
pole order along EIH (and zero divisor disjoint from the polar set). This proves
bb ) is
that the eect of the normalized blowing up of the fractional ideal OEIbb (EIH
bb
to replace the vertex X(I ? ) of EIbb by X(H \ I ? ). The strict transform of EIH
in this blow-up is clearly a Cartier divisor. If we do this for all the H 2 H that
contain I, then the vertex gets replaced by X(J) as asserted.
With Convention 2.3 in mind we write X j for X jH .
Proposition 5.3. For l su
ciently large, L(H;)(l) is generated by its sections.
We prove this proposition via a number of lemma's.
Let O be a ;-orbit in W(k) ; f0g. The projectivization of the hyperplane
perpendicular to w 2 O meets B precisely when (w w) < 0 and in that case
( w);1 de nes a meromorphic section of OB (;1). Let l be an integer > 2n + 1
and consider the Poincare-WeierstrassX
series
(l)
FO :=
( w);l :
Lemma 5.4. Let K w2O
be a compact subset. Then the set O0 of w
which P(w?) meets K is nite and the series
L
X
w2O;O0
2 O for
( w);l
converges uniformly and absolutely on K . In particular, FO(l) represents a meromorphic function on L that is homogeneous of degree ;l.
Proof. Since O is an orbit of the arithmetic group ; in W (k) ; f0g, the Ok -
submodule " W spanned by this orbit is discrete in W. Now O is also contained
in the real hypersurface a W de ned by (w w) = a for some a < 0. The
map m : a ! 0 1) de ned by m(w) := minz2K j(z w)j is proper. A standard
argument shows that the cardinality of " \ a \ m;1 (N ; 1 N] is proportional to
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE19
the (2n + 1)-volume of a \ m;1 (N ; 1 N] and hence is cN 2n for some c > 0.
So O0 is nite and for z 2 K we have the uniform estimate
X
w2O;O0
j(z w)j constant + X cN 2n(N ; 1);l :
1
N=2
The righthand side converges since l > 2n + 1.
For O and l as above, we extend FO(l) to each stratum L(I) of (L)tor by taking
the subseries de ning FO(l) whose terms makes sense on that stratum:
FO(l) (z) :=
X
w2O\I ?
(z w);l if z 2 L(I):
It is clear from the preceding that this subseries represents a meromorphic function
on L(I). The bundle L(I) ! B (I) is canonically isomorphic to the trivial bundle
with ber W=I ? ;f0g. So after choosing a generator of W=I ? we can think of this
subseries as de ning a rational function on B (I).
Lemma 5.5. The ;-invariant piecewise rational function on (L)tor dened by
FO(l) represents a meromorphic section of L over the toroidal compactication X tor
of X .
Proof. Given the k-isotropic line I W, choose a k-generator e 2 I and consider
the action of the right half plane on L de ned by Te (see Subsection 4.2): Tes (z) =
z + s(z e)e. In view of the characterization in Subsection 4.3 of the analytic
structure on (L)tor it is enough to show that limRe(s)!+1 FO(l) Tes jK exists as a
meromorphic function and this limit is equal to the subseries de ned above. We
claim that with K, a and " as in the proof of Lemma 5.4 above, the map
w) (z w) 2 K (" \ (a n I ? )) 7! Re (z(z
e)(e w)
is bounded from below, say by ;so . Once we establish this we are done, for then
w) + Re(s)
s 7! j(Tes z w)j = j(z e)(e w)j: (z(z
e)(e w)
is monotone increasing for Re(s) so for any for z 2 K and w 2 O n I ? , implying
that the limit is as stated.
In order to prove our claim, we rst note that without any loss of generality we
may assume that (z e) = 1 for all z 2 K (just replace z 2 K by (z e);1 z).
Write e0 for e and choose an isotropic e1 2 W(k) such that (e0 e1) = 1. Since the
orthogonal complement of C e0 + C e1 is negative de nite, we denote the restriction
of ; to this complement by h i. We write any w 2 W as w0e0 + w1e1 + w0 with
w0 ? (C e0 + C e1 ). Notice that the function w 2 " n I ? 7! jw1j = j(w e)j has
a positive minimum c > 0. Then for z 2 K and w 2 " \ (a n I ? ) we have the
20
EDUARD LOOIJENGA
estimate
w) = Re z0 w1 + w0 ; hz 0 w0i Re (z
(e w)
w1
;2
= Re(z0 + jw1j w0w1 ; hz 0 w0=w1i)
= Re(z0 ) + jw1j;2(a + hw0 w0i) ; hz 0 w0=w1 i)
= Re(z0 ) + jw1j;2a + kw0=w1 ; 12 z 0k2 ; 14 kz 0k2
Re(z0) ; c;2a ; 41 kz0k2:
The claim follows and with it, the lemma.
If I and O are such that O \ I ? is a single ;I -orbit, say of w0 2 I ? , then if H0
denotes the orthogonal complement of w0 in W, H0 \ I ? is independent of the choice
of w0. The same is true for the abelian divisor X(I)H0 in X(I) and the restriction of
FO(l) to the abelian variety X(I) is in fact the pull-back of a meromorphic function
on the elliptic curve X(I)=X(I)H0 = X(H0 \ I ? ). It has the Weierstrass form in
the sense of Lemma 5.6 below.
Lemma 5.6. Let L C be a lattice. Then for k 3 the series
X
}k (z) := (z + a);k
a 2L
represents a rational function on the elliptic curve C =L and for any k0 3, the }k
with k k0 generate the function eld of this curve.
We omit the proof of this well-known fact.
Proof of 5.3. It suces to prove this statement for L(H;)(k) ) instead of L(H;) for
some k > 0. We rst show that for k > 2n + 1 the sections of L(H;)(k) generate
this sheaf over X. Let H0 2 H and zo 2 H0 \ L. Choose w0 2 W(k) spanning the
orthogonal complement of H0 and let O denote the ;-orbit of w0. Then according
to Lemma 5.4 FO(k) de nes a section of L(H;)(k) . It is in fact a section of the
invertible subsheaf L(DHj 0 )(k) on X j and nonzero as such at z0 . These subsheafs
generate L(H;)(k) .
So it remains to verify the generating property over a cusp. But this follows
from the conjunction of Lemma's 5.5 and 5.6.
This has the following corollary, which we state as a theorem:
Theorem 5.7. The line bundle L and the collection of strict transforms of the
hypersurfaces DHbb in X jH satisfy the conditions of Theorem 3.9 so that the diagram
of birational morphisms and projective completions of X
X bb X jH X~ H ! X^ H
is dened. The morphism X~ H ! X bb is the blowup dened by OX bb (H;) on X bb.
The coherent pull-back of L(H;) to X~ H is semiample and denes the contraction
X~ H ! X^ H .
It is worth noting that the dierence X^ H ; X need not be a hypersurface: if
H is nonempty, then the complex codimension of this boundary is the minimal
dimension of a nonempty intersection of members of H in B . In many interesting
examples this is > 1 and in such cases a section of Lk over X extends to a section
of L^(H; )k . This gives the following useful application:
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE21
Corollary 5.8. Let H be an arithmetic arrangement on a complex ball B of di-
mension 2 that is arithmetic relative to the arithmetic group ;. Suppose that any
intersection of members of the arrangement H that meets B has dimension 2 in
B . Then the algebra of automorphic forms
k2Z
H 0 (B Oan(L)k);
(where L is the natural automorphic bundle over B and B is the arrangement
complement) is nitely generated with positive degree generators and its proj is the
modication X^ H of X bb .
6. Ball arrangements that are defined by automorphic forms
The hypotheses of Corollary 5.8 are in a sense opposite to those that are needed
to ensure that an arithmetically de ned ball arrangement is the zero set of an
automorphic form on that ball. This will follow from Lemma 3.7 and the subsequent
remark. Assume we are in the situation of this section and that we are given a
function H 2 H 7! nH 2 f1 2 g that is constant on the ;-orbits. This amounts
to giving an eective divisor on X bb supported by DHbb . Let I W be a k-isotropic
line contained in a member of H. Then the collection HI of H 2 H passing through
I decomposes into a nite collection of ;I -orbits of hyperplanes. Each H 2 HI
de nes a hyperplane (I ? =I)H in I ? =I with ;I -equivalent hyperplanes de ning the
same ;I -orbit.
P
Suppose now that H nH B H is the divisor of an automorphic form. Then
Lemma 3.7 implies that
X
nH (I ? =I)H
H 2;I nHI
should represent the Euler class of an ample line bundle. This Euler class is in
fact known to be the negative of the Hermitian form on I ? =I. This implies that
the intersection of the hyperplanes (I ? =I)H is reduced to the origin. In other
words: the collection of H 2 H containing a k-isotropic line is either empty or has
intersection equal to that line. Although this is formally weaker than the above
property, this simple requirement turns out be already rather strong.
P
Question 6.1. Suppose that the eective divisor H nH B H is at every cusp the
zero divisor of an automorphic function at that cusp. In other words, suppose
that the corresponding Weil divisor on the Baily-Borel compacti cation is in fact
a Cartier divisor. Is it then the divisor of an automorphic form?
P More generally, if
H 2 H 7! nH is Z-valued and ;-invariant in such a wayPthat H nH B H de nes a
Cartier divisor on the Baily-Borel compacti cation, is H nH B H then the divisor
of a meromorphic automorphic form?
In case such an automorphic form exists, we expect it of course to have a product
expansion.
7. Some applications
There are a number of concrete examples of moduli spaces to which Corollary
5.8 applies. To make the descent from the general to the special in Bourbakian
style, let us start out from the following situation: we are given an integral projective variety Y with ample line bundle and a reductive group G acting on the
pair (Y ). Assume also given a G-invariant open-dense subset U Y consisting
22
EDUARD LOOIJENGA
of G-stable orbits. Then G acts properly on U and the orbit space GnU exists
as a quasi-projective orbifold with the restriction jU descending to an orbifold
line bundle Gn(jU) over GnU. To be precise, geometric invariant theory tells us
that the algebra of invariants k2Z
H 0 (Y k )G is nitely generated with positive
degree generators. Its proj is denoted GnnY ss , because a point of this proj can
be interpreted as a minimal G-orbit in the semistable locus Y ss Y . It contains
GnU as an open dense subvariety. The orbifold line bundle Gn(jU) extends to an
orbifold line bundle Gnn over GnnY ss in a way that the space of global sections of
its kth tensor power can be identi ed with H 0(Y k )G .
Theorem 7.1. Suppose we are given identication of Gn(U jU) with a pair coming from a ball arrangement (X LjX ), such that
(i) any nonempty intersection of members of the arrangement H with B has dimension 2 and
(ii) the boundary GnnY ss ; GnU is of codimension 2 in GnnY ss .
Then this identication determines an isomorphism
k2Z
H 0(Y k )G H 0 (B Oan(L)k);
= k2Z
in particular, the algebra of automorphic forms is nitely generated with positive degree generators. Moreover, the isomorphism GnU = X extends to an isomorphism
GnnY ss = X^ H .
Proof. The codimension assumption implies that any section of Gn(k jU) extends
to (Gnnss )k . Since H 0(B O(L)k); = H 0(X Lk ), the rst assertion follows.
This induces an isomorphism between the underlying proj's and so we obtain an
isomorphism GnnY ss = X^ H , as stated.
The identi cation demanded by the theorem will usually come from a period
mapping. In such cases the codimension hypotheses of are often ful lled. Let us
now be more concrete.
7.1. Unitary lattices attached to directed graphs. The only imaginary quadratic elds we will be concerned with are the cyclotomic ones, i.e., Q(4) and
Q(6). Their rings of integers are the Gaussian integers Z4] and the Eisenstein
integers Z6] respectively. It is then convenient to have the following notation at
our disposal. Let D be a nite graph without loops and multiple edges and suppose
all edges are directed, in other words D is a nite set I plus a collection of 2-element
subsets of I, each such subset being given as an ordered pair. Then a Hermitian
lattice Zk]D is de ned for k = 4 6 as follows: Zk]D is the free Zk]-module on
the set of vertices (ri)i2I of D and a Zk]-valued Hermitian form on this module
de ned by
8
>
<j1 + k j2 = k=2 if j = i
(ri rj ) = >;1 ; k
if (i j) is a directed edge
:0
if i and j are not connected:
We denote the group of unitary transformations of Zk]D by U(D k ). This is
an arithmetic group in the unitary group of the complexi cation of Zk]D . The
elements in Zk]D of square norm k=2 are called the roots of Zk]D .
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE23
If D is a forest, then the isomorphism class Zk]D is independent of the way the
edges are directed. Usually it is only the isomorphism class that matters to us, and
so in this case there is no need to specify these orientations.
7.2. The moduli space of quartic curves. It is well-known fact that the anticanonical map of a Del Pezzo surface of degree two realizes that surface as a double
cover of a projective plane rami ed along a smooth quartic curve and that this
identi es the coarse moduli space of Del Pezzo surfaces of degree two and that of
smooth quartic curves. The latter is also the the coarse moduli space of nonhyperelliptic genus three curves. The invariant theory of quartic curves is classical
(Hilbert-Mumford). A period map taking values in a ball quotient has been constructed by Kondo 9]. Heckman recently observed that the relation between the
two is covered by Theorem 7.1 (unpublished) and that the situation is very similar
to the case of rational elliptic surfaces discussed below, it is only simpler. We briey
review this work.
Let us rst recall the invariant theory of quartic curves. Fix a projective plane
P. Put Hk := H 0 (P OP ), so that Pk := P(Hk) is the space of eective degree k
divisors on P . We take Y := P4 and := OY (1) and G := SL(H1). Following 15]
a point of P4 is G-stable precisely if the corresponding divisor is reduced and has
cusp singularities at worst (i.e., locally formally given by y2 = xi with i = 1 2 3).
It is semistable precisely when it is reduced and has tacnodes at worst (like y2 = xi
with 1 i 4) or is a double nonsingular conic. A divisor has a minimal strictly
semistable orbit precisely when it is sum of two reduced conics, at least one of which
is nonsingular, which are tangent to each other at two points. If (x y) are ane
coordinates in P, then these orbits are represented by the quartics (xy ;0 )(xy ;1 ),
with 0 : 1 ] 2 P1 and so GnnY ss ; GnY st is a rational curve. The nonsingular
quartics are stable and de ne an open subset Y 0 Y .
Kondo's period map is de ned as follows: given a smooth quartic curve Q P,
he considers the 4-cover of P that is totally rami ed along Q. This is a smooth
K3-surface of degree 4 that clearly double covers the degree two Del Pezzo surface
attached to P. For such surfaces there is a period map taking values in an arithmetic
group quotient of a type IV domain of dimension 19. But the 4-symmetry makes
it actually map in a ball quotient of dimension 6. To be precise, consider the
Hermitian lattice Z4]E7 . Its signature is (6 1) and so its complexi cation de nes
a 6-dimensional ball B . We form X := U(E7 4)nB . It turns out that U(E7 4)
acts transitively on the set of cusps so that X bb is topologically the one-point
compacti cation of X. The roots (i.e., the vectors of square norm 2) come in two
U(E7 4 )-equivalence classes: the one represented by the orthogonal complement
of any generator ri , denoted Hn , and the rest, Hh , represented by ri ; (1 + 4)rj ,
where ri rj are two generators with (ri rj ) = 1 + 4. This de nes a hypersurface
D in X with two irreducible components Dn and Dh respectively. The hyperplane
sections of B of type Hh are disjoint and so Dh has no self-intersection. Kondo's
theorem states that the period map de nes isomorphisms
GnY 0 = X ; D GnY st = X ; Dh :
The last isomorphism is covered by an isomorphism of orbiline bundles Gnn !
LjX ; Dh and so Theorem 7.1 applies with U = Y ss: we nd that the period map
extends to an isomorphism
GnnY ss = X^ Hh :
24
EDUARD LOOIJENGA
7.3. A nine dimensional ball quotient. We begin with a Deligne-Mostow example which involves work of Allcock 1]. Fix a projective line P , put Hk :=
H 0(P OP (k)), and identify Pk := P(Hk) with the k-fold symmetric product of P,
or what amounts to the same, the linear system of eective degree k divisors on P.
The group G := SL(H1) acts on the pair (Pk OPk (1)). Following Hilbert, a degree
k divisor is stable resp. semistable if and only if all its multiplicities are < k=2
resp. k=2. We only have a strictly semistable orbit if k is even 4 and in that
case there is a unique minimal such orbit: the divisors with two distinct points of
multiplicity k=2. The nonreduced divisors de ne a discriminant hypersurface in
Pk , whose complement we shall denote by Pk0 .
Now take k = 12. Given a reduced degree 12 divisor D on P we can form
the 6 -cover C ! P that is totally rami ed over D. The Jacobian J(C) of C
comes with an action of the covering group 6 and the part where that group acts
with a primitive character de nes a quotient abelian variety of dimension 10. The
isomorphism type of this quotient abelian variety is naturally given by a point in
a ball quotient. To be precise, if A10 is the string with 10 nodes, then Z6]A10 is
nondegenerate of signature (9 1) and so de nes a ball B of complex dimension 9.
The group U(A10 6 ) is arithmetic and so we may form
X := U(A10 6 )nB :
It acts transitively on the cusps so that the Baily-Borel compacti cation X bb of X is
a one-point compacti cation. The hyperplanes perpendicular to a root makes up an
arrangement that is arithmetically de ned. Since they are transivily permuted by
U(A10 6 ), they de ne an irreducible hypersurface D in X. A theorem of DeligneMostow 4] implies that X ; D parametrizes the isomorphism types of the quotients
of the Jacobians that we encounter in the situation described above and that the
`period map'
GnP120 ! X ; D
is an isomorphism. It also follows from their work (see also 7]) that this period
isomorphism extends to isomorphisms
GnP12st = X GnnP12ss = X bb
(with the unique minimal strictly semistable orbit corresponding to the unique
cusp) and that the latter identi es L12 with ;nOP12ss (1). (In this situation ; is not
neat and so L is merely an orbiline bundle on X bb .)
We can also think of GnP120 as the moduli space of hyperelliptic curves of genus
5 or as the moduli space of 12-pointed smooth rational curves (with the 12 points
not numbered). For either interpretation this moduli space has a Deligne-Mumford
compacti cation and these two coincide as varieties. It can be veri ed that the
Deligne-Mumford compacti cation is just the minimal normal crossings blowup in
the sense of Subsection 2.3 of D in X bb .
7.4. Miranda's moduli space of rational elliptic surfaces. In this discussion
a rational elliptic surface is always assumed to have a section. The automorphisms
of such a surface permute the sections transivitively, so when we are concerned with
isomorphism classes, there is no need to single out a speci c section. Contraction
of a section yields a Del Pezzo surface of degree one and vice versa, so the moduli
space of rational elliptic surfaces is the same as the moduli space of degree one Del
Pezzo surfaces. A rational elliptic surface can always be represented in Weierstrass
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE25
form: y2 = x3 + 3f0 (t)x + 2f1 (t), where x y t are the ane coordinates of a P2bundle over P1 and f0 and f1 are rational functions of degree 4 and 6 repectively.
(We normalized the coecients as to have the discriminant of the surface take the
simple form f03 +f12 .) This is Miranda's point of departure 14] for the construction
of a compacti cation by means of geometric invariant theory of this moduli space.
With the notation of the previous subsection, let Y be the `orbiprojective' space
obtained as the orbit space of H4 H6 ; f(0 0)g relative to the action of the
center C GL(H1). It comes with a natural ample orbifold line bundle over Y
endowed with an action of G = SL(H1). It has the property that the pull-back of
OP12 under the equivariant `discriminant morphism'
: Y ! P12 (f0 f1 ) 7! f03 + f12
is equivariantly isomorphic to 6 . The above morphism is nite and birational
with image a hypersurface. The preimage ;1(P120 ), which clearly parametrizes the
rational elliptic brations over P with reduced discriminant, maps isomorphically
to its image in P12 and is contained in the Y st . We denote its G-orbit space by M.
But a stable point of Y need not have a stable image in P12: Miranda shows that
the G-orbit space of Y st \ ;1(P12st ), denoted here by M~, parametrizes the Mirandastable rational elliptic surfaces with reduced bers and stable discriminant, in other
words, the allowed singular bers have Kodaira type Ik with k < 6, II, III or IV ,
whereas the dierence GnY st ; M~parametrizes rational elliptic surfaces with an
Ik - ber with 6 k 9. The minimal orbits in Y ss ; Y st parametrize rational elliptic surfaces with two I0 - bers or a I4 - ber (they make up a rational curve). So the
projective variety MM := GnnY ss still parametrizes distinct isomorphism classes of
elliptic surfaces. We regard this variety as a projective completion of M~and call
it the Miranda compactication. The boundary MM ; M~is of codimension 5 in
MM and MM ; M is a hypersurface in MM .
7.5. A period map for rational elliptic surfaces. Assigning to a rational elliptic bration with reduced discriminant its discriminant de nes a morphism
M = GnY 0 ! GnP120 ! X ; D:
and the pull-back of L12 can be identi ed with Gn6. It is proved by HeckmanLooijenga 7] that this morphism is a closed embedding with image a deleted hyperball quotient. To be precise, choose a vector in Z6]A10 of square norm 6 (for instance, the sum of two perpendicular roots) and let "o Z6]A10 be its orthogonal
complement. It is shown in op. cit. that all such vectors are U(A10 6)-equivalent
and for that reason the choice of this vector is immaterial for what follows. The
U(A10 6 )-stabilizer of "o acts on "o through its full unitary group, which we denote
by ;o . The sublattice "o de nes a hyperball B o B , a ball quotient Xo := B o;o
and a natural map p : Xo ! X. The restriction of H to B o is an arrangement on
B o that is arithmetically de ned and the corresponding hypersurface Do on Xo is
just p;1(D). It is proved in 7] that the period map de nes an isomorphism
M = GnY 0 = Xo ; Do :
It is easily seen to extend to a morphism M~! Xo . This extension is not surjective.
This is related to the fact that Do has four irreducible components, or equivalently,
that the restriction of the arrangement H to C O "o is not a single ;o -equivalence
class, but decomposes into four such classes. These four classes are distinguished
26
EDUARD LOOIJENGA
by a numerical invariant (for the de nition of which we refer to op. cit.) that takes
the values 6 9 15 18. So HjC O "o is the disjoint union of the ;o -equivalence
classes H(d) with d running over these four numbers and H(d) de nes an irreducible
hypersurface D(d) Xo . The image of the period map is the complement of
D(6) D(9). In fact, we have an isomorphism
M~= Xo ; (D(6) D(9))
with the generic point of D(18) resp. D(12) describing semistable elliptic brations
with a I2 resp. II- ber. This isomorphism is covered by an isomorphism of orbifold
line bundles: Gn6 is isomorphic to the pull-back of the L12. It is shown in 7] that
any nonempty intersection of hyperplane sections of B o taken from H(6) H(9) has
dimension 5. Since MM ; M~is of codimension 5 on MM , theorem 7.1 applies
with U := Y st \ ;1(P12st ) and we nd:
Corollary 7.2. The period map extends to an isomorphism of the Miranda moduli
space MM of rational elliptic surfaces onto the modication X^oH(6)H(9) of Xobb
dened by the arrangement H(6) H(9).
7.6. The moduli spaces of cubic surfaces and cubic threefolds. The previous two examples describe the moduli spaces of Del Pezzo surfaces of degree one
and two as a ball quotient and express its GIT compacti cation as a modi ed BailyBorel compacti cation of the ball quotient. This can also be done for Del Pezzo
surfaces of degree three, that is for smooth cubic surfaces. Allcock, Carlson and
Toledo gave in 3] a 4-ball quotient description of this moduli space, by assiging to
a smooth cubic surface S P3 rst the 3 -cover K ! P3 totally rami ed over S
and then the intermediate Jacobian of K with its 3 -action. But in this case they
nd that the GIT compacti cation coincides with the Baily-Borel compacti cation
(one might say that the relevant arrangement is empty). However, they recently
extended this to the moduli space of smooth cubic threefolds: the GIT compacti cation has been recently given by Allcock 2], whereas Allcock, Carlson and Toledo
found a 10-ball quotient description of the moduli space (yet unpublished). It comes
naturally with an arrangement and it seems that Theorem 7.1 applies here, too.
References
1] D. Allcock: The Leech Lattice and Complex Hyperbolic Reections, Invent. Math. 140 (2000),
283{301.
2] D. Allcock: The moduli space of cubic threefolds, to appear in J. Alg. Geom., also available
at http://www.math.harvard.edu/ allcock/research/3foldgit.ps.
3] D. Allcock, J.A. Carlson, D. Toledo: The complex hyperbolic geometry for moduli of cubic
surfaces, 55 pp., available at math.AG/0007048, to appear in J. Alg. Geom., see also A complex
hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris, t. 326, Serie I (1998),
49{54.
4] P. Deligne, G.D. Mostow: Monodromy of hypergeometric functions and non-lattice integral
monodromy, Inst. Hautes E tudes Sci. Publ. Math. 63 (1986), 58{89.
ements de geometrie algebrique. II. Etude
5] A. Grothendieck: El
globale elementaire de quelques
classes de morphismes. Inst. Hautes E tudes Sci. Publ. Math. 8 (1961), 222 pp.
6] R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag,
New York-Heidelberg (1977).
7] G. Heckman, E. Looijenga: The moduli space of rational elliptic surfaces, 47 pp., to appear
in the Proceedings of the Conference Algebraic Geometry 2000 (Azumino), also available at
math.AG/0010020.
8] Yi Hu: A Compacti cation of Open Varieties, 21 pp., available at math.AG/9910181.
COMPACTIFICATIONS DEFINED BY ARRANGEMENTS I: THE BALL QUOTIENT CASE27
9] S. Kondo: A complex hyperbolic structure on the moduli space of curves of genus three,
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Faculteit Wiskunde en Informatica, Universiteit Utrecht, Postbus 80.010, NL-3508
TA Utrecht, Nederland
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